Evaluating the performance of memory type logarithmic estimators using simple random sampling

In survey research, various types of estimators have been suggested that consider only the current sample information to compute the unknown population parameters. Therefore, we utilize the past sample information along with the current sample information in the form of hybrid exponentially weighted moving averages to suggest the memory type logarithmic estimators for time-based surveys. The expression of the mean square error of the suggested estimators is determined to the first order of approximation. A relative comparison of the suggested estimators with the existing estimators is performed and efficiency conditions are obtained. Further, a simulation study is accomplished using a hypothetically rendered population and a real data illustration to improve the theoretical results. The results of the simulation study and the real data application exhibit that the consideration of past and current sample information meliorates the efficiency of the suggested estimators.


Introduction
In sample surveys, usually, the information on an auxiliary variable, namely, population mean � X, population standard deviation S x , population correlation coefficient ρ xy , population coefficient of skewness β 1 (x), population coefficient of kurtosis β 2 (x) etc., are known in advance which are positively or negatively correlated with the study variable. The utilization of this information helps to enhance the efficiency of the suggested estimators. [1] introduced the ratio estimator when the auxiliary information is positively correlated with the study variable as  [2] investigated the classical product estimator when the auxiliary information is negatively correlated with the study variable as where � x ¼ n À 1 P n i¼1 x i and � y ¼ n À 1 P n i¼1 y i are the sample means of the auxiliary variable x and study variable y respectively.
The mean square error (MSE) of the above ratio and product estimators is given as MSEð� where f = 1/n, � Y is the population mean of study variable y, C x and C y are the population coefficient of variations of the auxiliary and study variables, respectively.
[3] suggested a class of logarithmic type estimator as where α is a duly opted scalar.
The MSE of the estimator � y bk up to the first order of approximation is given by The minimum MSE at optimum value of α (opt) = −ρ xy (C y /C x ) is given by Over the recent years, utilizing the current sample information, a large number of ratio, product, regression and exponential type estimators consist of different auxiliary information have been suggested by several authors. [4] utilized the known correlation coefficient for estimating the population mean. [5] suggested a general estimation procedure of population mean using known values of population parameters. [6] introduced estimation of entropy with application to test of fit under RSS. [7] suggested the estimation of proportion in presence of tie using RSS. [8] discussed interval estimation of P(X < Y) using RSS. [9] suggested estimation of a symmetric distribution function in multistage RSS, whereas [10] examined a new goodness of fit tests for the Cauchy distribution. [11] suggested improved modified ratio estimators using robust regression estimators. [12] suggested some efficient classes of estimators under stratified random sampling, whereas [13] suggested some novel class of estimators of population mean under ranked set sampling (RSS). [14,15] developed some log type class of population mean estimators under RSS. [16] suggested quantile regression ratio type estimators of population mean using complete and partial auxiliary information, whereas [17] proposed quantile robust regression type estimators based on minimum covariance determinant for estimating population mean. [18] suggested two auxiliary variable based robust regression ratio type estimators of the population mean. [19] analyzed the covid-19 risk using an exponential estimators. [20] developed double sampling based robust ratio estimators in the presence of outliers. [21] considered an improved class of robust ratio estimators by using the minimum covariance determinant estimation. [22] investigated robust regression type estimators to estimate population mean using simple and two-stage sampling. [23] introduced robust ratio type estimators for estimating population mean under simple random sampling. [24] examined the Poisson regression ratio estimators for population mean using double sampling. [25] proposed compromised imputation based mean estimators using robust quantile regression method. It has been well established that the consideration of auxiliary information helps to meliorate the efficacy of the estimators. Due to this reason, only the current sample � y t at time t is utilized to obtain the knowledge of the study and auxiliary variables. The efficiency of the estimator can further be enhanced by utilizing the knowledge of the current and the previous samples such as � y tÀ 1 from time t − 1, � y tÀ 2 À 1 from time t − 2 and all this. It becomes essential in situations when a particular time interval based survey such as monthly or annual is performed. [26] introduced memory type estimators for population mean utilizing exponentially weighted moving averages (EWMA) for time scaled surveys under SRS, whereas [27] developed the memory type product and ratio estimators for population mean utilizing hybrid exponentially weighted moving averages (HEWMA) for time-based surveys under SRS. Later on, [28] suggested memory type product and ratio estimators of population mean in stratified sampling, whereas [29] investigated the memory type product and ratio estimators of population mean utilizing ranked-based sampling schemes. [30] studied a memory type shrinkage estimator of population mean in the quality control process. [31] suggested the ratio and product type estimators for estimating population mean for time-based survey. Recently, [32] suggested memory-type ratio and product estimators for population variance utilizing EWMA statistics for time-scaled surveys.
In the theory of survey sampling, the classical estimators are based on functional forms, like chains, exponential functions, and linear combinations. However, the construction of estimators using the logarithmic function is novel. Development of advance data analysis tools has enhanced the computational limitations of these functions. In the real world, logarithms are utilized for the measurement of earthquakes on Richter scales, acidity on pH scale, and sound on decibels. In addition, logarithms can also be utilized to measure exponential decay and growth, like interest rates, bacterial growth in a Petri dish, and radioactive decay in carbon dating. Some real life examples are also given in Section 6 and a numerical study is performed. Therefore, it is important to study logarithmic type estimators for the estimation of population parameters and this is the motivation behind the construction of the suggested class of estimators. This study aims to proffer a memory type logarithmic estimator based on the knowledge of the past as well as current sample information. To achieve this aim, the HEWMA statistic is considered in this study.
The plan of this article is described in few sections. In Section 2, HEWMA statistic is described along with the review of the existing memory type estimators along with their properties. The developed memory type logarithmic estimator is given in Section 3 along with its properties. In Section 4, a comparative study is performed by comparing the MSE expressions of the suggested and existing estimators. A simulation study is accomplished in Section 5 and a real data illustration is given in Section 6. The conclusion is outlined in Section 7.
2 Reviews of estimators based on HEWMA statistic [33,34], respectively, introduced the concept of the EWMA statistic and the HEWMA statistic for efficiently detecting the changes in the fundamental process mean in control charting. Let X 1 , X 2 , . . .., X n be an independent and identically distributed random variables from which the sequence HE 1 , HE 2 , . . ., HE n are defined by utilizing the undermentioned recurrence expressions.
where λ i , i = 1, 2 are duly chosen scalars. Also, E t and HE t denote the EWMA and HEWMA statistic, respectively. The initial values of these statistics are considered as expected mean which can be estimated from a pilot survey. In this study, it is considered as zero i.e. HE 0 = Et 0 = 0. The mean and variance of HE t statistic were computed by [34]. Later on, [35] observed that the expressions of mean and variance derived by [34] were incorrect, then [35] provided the correct expressions of mean and variance of HE t statistic as where t � 1, � Y and s 2 y are the mean and variance of the study variable. The limiting form of the variance is expressed by [27] considered the HEWMA statistic to propose the memory type ratio and product estimators. The HEWMA statistic for the study variable is given by and the HEWMA statistic for the auxiliary variable is given by The statistic Z t and Q t are unbiased for population mean � Y and � X respectively. Utilizing the statistic Z t and Q t , [27] suggested the memory type ratio and product estimators under simple random sampling as To deduce the properties of the suggested estimators, we consider the following annotations.
The MSE of the ratio and product estimators to the first order of approximation are given by It is to be noted that the value of δ will be replaced by δ 1 which is given by

Suggested memory type logarithmic estimators
The logarithmic function has a few important characteristics and furnishes an essential role in various dimensions of scientific and non-scientific disciplines. In this article, we have developed the logarithmic relationship between the study variable Y and the auxiliary variable X for computing the population mean � Y . The developed estimators would perform in situations where the variables Y and X are logarithmically associated. Motivated by the works of [26,27], we propose the memory type logarithmic estimator using the HEWMA statistic as where β is a duly opted scalar.
Using the notations described in the preceding section, we rewrite the estimator � y M bk in terms of e 0 s as Squaring and taking expectation both sides of (24), we get the MSE of the estimator � y M bk to the first order of approximation as Minimizing the MSEð� y M bk Þ with respect to β, we get The minimum MSE at the optimum value of β is given by The above MSE expression is important to perform the comparative study in the next section.

Comparative study
To derive the efficiency conditions, we perform a comparative study by comparing the minimum MSE of the developed memory type logarithmic estimator with the minimum MSE of the existing conventional and memory type estimators.
1. By comparing the minimum MSE of the developed memory type logarithmic estimator from (27) with the minimum MSE of the usual mean estimator, we get 2. By comparing the minimum MSE of the developed memory type logarithmic estimator from (27) with the minimum MSE of the classical ratio estimator from (3), we get 3. By comparing the minimum MSE of the developed memory type logarithmic estimator from (27) with the minimum MSE of the classical product estimator from (4), we get 4. By comparing the minimum MSE of the developed memory type logarithmic estimator from (27) with the minimum MSE of the conventional logarithmic type estimator from (7), we get 5. By comparing the minimum MSE of the developed memory type logarithmic estimator from (27) with the minimum MSE of the memory type ratio estimator from (20), we get 6. By comparing the minimum MSE of the developed memory type logarithmic estimator from (27) with the minimum MSE of the memory type product estimator from (21), we get Under the above efficiency conditions, the suggested memory type logarithmic estimator dominates the existing conventional and memory type estimators. Furthermore, these conditions are substantiated with a simulated study and a real data illustration.

Simulation study
To illustrate the effectiveness of the suggested memory type logarithmic estimators against the existing conventional and memory type ratio and product estimators, we conduct a simulation study over an artificially rendered bivariate normal population of size N = 1000 units using R software with parameters � X ¼ 40, � Y ¼ 30, σ x = 6 and σ y = 5. Using 40,000 iterations, the MSE and relative efficiency (RE) of various estimators (t) are computed by utilizing the undermentioned expressions: The performance of the developed memory type logarithmic estimator is studied by considering various amounts of correlation coefficients ρ xy = ±0.  values of ρ xy , n and λ 2 . From the simulation results, we draw the interpretation in point-wise fashion as follows: 1. From Tables 1 and 2, it has been observed that the suggested memory type logarithmic estimator � y M bk dominates the usual mean estimator � y, classical ratio estimator � y r , logarithmic estimator � y bk , memory type ratio estimator � y M r in terms of lesser MSE and greater RE for different amount of ρ xy = +0.1, +0.3, +0.5, +0.7, +0.9. Tables 3 and 4, it has been observed that the suggested memory type logarithmic estimator � y M bk dominates the usual mean estimator � y, classical product estimator � y p , logarithmic estimator � y bk , memory type product estimator � y M p in terms of lesser MSE and greater RE for different amount of ρ xy = −0.1, −0.3, −0.5, −0.7, −0.9. Table 1, it has been observed that the MSE of the suggested memory type logarithmic estimator � y M bk decreases as the values of λ 2 increase, whereas a reverse inclination can be seen from the results of Table 2 reported in terms of RE. 4. The similar inclination can be seen from the results of Tables 3 and 4 reported, respectively, in terms of MSE and RE. Tables 1 and 3, it is noticed that the MSE of the various estimators decreases as the sample size increase for each value of ρ xy , whereas from Tables 2 and 4, it is seen that the RE increases as the sample size increases for each value of ρ xy . Tables 1 and 3, it is also observed that the MSE of various estimators decreases as the values of correlation coefficient increases, whereas from Tables 2 and 4, it is seen that the RE increases as the values of ρ xy increases.

Real data illustration
In the present section, we exemplify the suggested estimators using a real data from [36], pp. 31. The real data is based on the density as the study variable Y and stiffness as the 1. We select 25 samples each of size 5 from the above discussed population using simple random sampling without replacement, which are listed in Table 5 for ready reference.
2. The usual mean estimator � y, usual ratio estimator � y r , logarithmic type estimator � y bk , memory type ratio estimator � y M r and suggested memory type logarithmic estimator � y M bk are computed using their respective expressions.
3. Using λ 1 = 0.10 and λ 2 = 0.25, the HEWMA statistic is also tabulated for each sample and the time-varying MSE for different estimators is also listed in Table 5. 4. The MSE for the memory type ratio estimator � y M r and the suggested memory type estimator � y M bk is computed as 0.0692 and 0.0109, respectively, with the help of expressions (20) and  (27) which shows that the suggested memory type logarithmic estimator dominates the existing estimators.

Conclusion
In this article, we have developed a memory type logarithmic estimator by incorporating the past sample information along with the current sample information in the form of HEWMA statistic. The mathematical property, namely, MSE of the developed memory type logarithmic estimators is obtained to the first order of approximation. Further, a simulation study is performed over an artificially generated population and presented the simulation results from Tables 1 to 4 that confirm the dominance of the suggested memory type logarithmic estimators over the usual mean estimator, classical ratio and product estimators, the conventional logarithmic estimator envisaged by [3] and memory type ratio and product estimators suggested by [27]. Moreover, a real data illustration is also presented in support of the theoretical results and the results are reported in Table 5 that also show the dominance of the suggested estimators over their entrants. Therefore, the suggested memory type logarithmic estimator is enthusiastically recommended to the survey practitioners for time-based surveys.
In the forthcoming article, we intend to develop the suggested memory type logarithmic estimator using the HEWMA statistic under stratified random sampling and ranked set sampling.